Semiparametric Proportional Hazards Model Research Articles

Choice of Parametric Models in Survival Analysis: Applications to Monotherapy for Epilepsy and Cerebral Palsy

Summary In the analysis of medical survival data, semiparametric proportional hazards models are widely used. When the proportional hazards assumption is not tenable, these models will not be suitable. Other models for covariate effects can be useful. In particular, we consider accelerated life models, in which the effect of covariates is to scale the quantiles of the base-line distribution. Solomon and Hutton have suggested that there is some robustness to misspecification of survival regression models. They showed that the relative importance of covariates is preserved under misspecification with assumptions of small coefficients and orthogonal transformation of covariates. We elucidate these results by applications to data from five trials which compare two common anti-epileptic drugs (carbamazepine versus sodium valporate monotherapy for epilepsy) and to survival of a cohort of people with cerebral palsy. Results on the robustness against model misspecification depend on the assumptions of small coefficients and on the underlying distribution of the data. These results hold in cerebral palsy but do not hold in epilepsy data which have early high hazard rates. The orthogonality of coefficients is not important. However, the choice of model is important for an estimation of the magnitude of effects, particularly if the base-line shape parameter indicates high initial hazard rates.

A Bayesian Semiparametric Analysis of the Reliability and Maintenance of Machine Tools

A Bayesian semiparametric proportional hazards model is presented to describe the failure behavior of machine tools. The semiparametric setup is introduced using a mixture of Dirichlet processes prior. A Bayesian analysis is performed on real machine tool failure data using the semiparametric setup, and development of optimal replacement strategies are discussed. The results of the semiparametric analysis and the replacement policies are compared with those under a parametric model.

Survival and Default of Original Issue High-Yield Bonds

We study the default behavior of original issue high-yield bonds to answer the open question of how the probability of default changes over time. We use a flexible econometric method, the Cox proportional hazard, to model the default behavior of junk bonds over their life. The method allows us to include the impact of issue and issuer characteristics on the probability of and the time to, default in the estimation. Using a large comprehensive sample, we find that the bonds face a constantly increasing default risk over time, with the most significant increase beyond four years after issuance. Due to the higher costs of high-yield bonds, issuers will refinance their debt as their prospects improve. Only nonperforming issuers will keep their junk bonds outstanding over the medium to long run. Therefore, the default rate of high-yield bonds should exhibit an increasing pattern over time. In this paper, we suggest a new application for a semi-parametric proportional hazard model. This new application allows us to extract default probabilities of original-issue high yield (junk) bonds. This method largely avoids the estimation problems caused by the changing population of outstanding junk bonds, and we can use it to account for the impact of issue and issuer characteristics on the probability of, and the time to, default in the estimation. In contrast to earlier studies, we find strong evidence for default probabilities that increase monotonically as junk bonds age, with the most significant increase coming four years after issuance. We analyze several issue-years together without distortions through changing populations. To do so, we first identify several characteristics at the issue date that have an impact on the default of bonds. We use these variables to estimate a semi-parametric Cox proportional hazard model. Using the coefficient estimates, we can recover the baseline hazard functions and the instantaneous default probabilities over time. We also estimate a time-varying Cox proportional hazard model, using the time-varying spread between interest rates at the bond's issue date and current interest rates as an explanatory variable. This specification removes the impact of the general level of interest rates from our hazard estimations. The fact that we also observe the increasing risk of default in this setup suggests that our results would still hold in a period of stable interest rates. As junk bonds age, the increasing risk of default can be explained by the nature of this method of corporate financing. As with any kind of borrowing, higher-risk borrowers have to

Widowhood-related mortality in Scania, Sweden during the 19th century

In this article, mortality risks are empirically examined in connection with spousal bereavement in four parishes in Scania, southern Sweden, during the 19th century using duration analysis. The data are longitudinal and the inhabitants have been followed literally from the cradle to the grave. To simultaneously catch transitory (shock) and persistent (long-term) effects of widowhood on mortality of the surviving spouse, the Cox semiparametric proportional hazards model has been applied with time spent in widowhood as a time-dependent multiple factor. Widowers in general were found to face higher relative mortality risks than widows and the effect of bereavement decreased through time. The estimated relative risks for males were dependent on socioeconomic status and widowers classified as landless faced high relative risks. Quantitatively, the magnitude of our estimates was large in comparison with similar studies made on contemporary data.

Exact tests for exponential regression

Abstract The exponential distribution is a fundamental model in parametric survival analysis. In this paper the exponential regression model with a canonical parameterization, λ=β t x , is explored. Large sample and exact significance tests are developed to test the relationship between survival and one covariate. The standard unconditional likelihood approach is compared with two new tests, which are based on the conditional distribution of the sufficient statistics. The various testing procedures are compared numerically. Finally, the relationship with the semiparametric proportional hazard model is investigated.

A Proportional Hazards Model for the Subdistribution of a Competing Risk

With explanatory covariates, the standard analysis for competing risks data involves modeling the cause-specific hazard functions via a proportional hazards assumption. Unfortunately, the cause-specific hazard function does not have a direct interpretation in terms of survival probabilities for the particular failure type. In recent years many clinicians have begun using the cumulative incidence function, the marginal failure probabilities for a particular cause, which is intuitively appealing and more easily explained to the nonstatistician. The cumulative incidence is especially relevant in cost-effectiveness analyses in which the survival probabilities are needed to determine treatment utility. Previously, authors have considered methods for combining estimates of the cause-specific hazard functions under the proportional hazards formulation. However, these methods do not allow the analyst to directly assess the effect of a covariate on the marginal probability function. In this article we propose a novel semiparametric proportional hazards model for the subdistribution. Using the partial likelihood principle and weighting techniques, we derive estimation and inference procedures for the finite-dimensional regression parameter under a variety of censoring scenarios. We give a uniformly consistent estimator for the predicted cumulative incidence for an individual with certain covariates; confidence intervals and bands can be obtained analytically or with an easy-to-implement simulation technique. To contrast the two approaches, we analyze a dataset from a breast cancer clinical trial under both models.

Is Workers' Compensation a Substitute for Unemployment Insurance?

This paper examines how the Workers' Compensation (WC) and Unemployment Insurance (UI) programs interact to influence the duration of claims due to workplace accidents. We use longitudinal WC administrative micro-data on more than 30,000 workers in the Canadian construction industry for the period 1976–1986. For the estimations, we use the Meyer (1990) semi-parametric proportional hazard model. Our results show, in particular, that a reduction in the UI replacement ratio is associated with an increase in the duration of claims due to severe accidents that are difficult to diagnose. Moreover, the duration of spells on WC is much higher when an accident occurs in December, a month which corresponds to the beginning of the lay-off season in the construction sector. This result is consistent with the fact that WC benefits are more generous than UI benefits in Canada.

Statistical Inference of a Bivariate Proportional Hazard Model with Grouped Data

This paper studies the estimation of a semiparametric bivariate proportional hazard model from grouped event time data. As a direct generalization of the bivariate exponential distribution of Marshall and Olkin, the model, on the one hand, controls for the effects of observed covariates, and on the other, achieves great exibility through nonparametrically specified baseline hazards. The model is most relevant in analyzing the joint distribution of two event times arising from systems of two components. Examples include the two infection times of the left and the right kidneys of patients and the two retirement times of married couples. To estimate this semiparametric model from grouped data, we propose a maximum likelihood estimator and a minimum chi-square estimator. Both estimation methods exploit the fact that the most flexible model structure that can be identified with grouped data is finite-dimensional. Compared with the maximum likelihood estimation, the minimum chi-square procedure is computationally more attractive but applies only to many observations per cell cases where the covariates are either categorical or amendable to sensible grouping. Specification tests for different model assumptions are also discussed.

A Generalized Moments Specification Test of the Proportional Hazards Model

Abstract This article describes a generalized moments specification test of the semiparametric proportional hazards model of Cox. In contrast to other specification tests for this model, the generalized moments test is applicable in the presence of continuous explanatory variables and does not require assigning the data to predetermined cells. The results of a Monte Carlo investigation suggest that the generalized moments test has good finite-sample properties and that it is more powerful than other available tests for models with continuous explanatory variables when the alternative is an accelerated failure time model. The test is based on generalized residuals for the proportional hazards model. When the model is correct, the generalized residuals have asymptotically the (possibly censored) unit exponential distribution. The generalized moments are means of functions of the generalized residuals. The generalized moments have known values when the generalized residuals have the unit exponential distribu...

A Generalized Moments Specification Test of the Proportional Hazards Model

Abstract This article describes a generalized moments specification test of the semiparametric proportional hazards model of Cox. In contrast to other specification tests for this model, the generalized moments test is applicable in the presence of continuous explanatory variables and does not require assigning the data to predetermined cells. The results of a Monte Carlo investigation suggest that the generalized moments test has good finite-sample properties and that it is more powerful than other available tests for models with continuous explanatory variables when the alternative is an accelerated failure time model. The test is based on generalized residuals for the proportional hazards model. When the model is correct, the generalized residuals have asymptotically the (possibly censored) unit exponential distribution. The generalized moments are means of functions of the generalized residuals. The generalized moments have known values when the generalized residuals have the unit exponential distribution, and an asymptotic χ 2 statistic tests for whether the differences between sample analogs of the generalized moments and the known values are larger than can be explained by random sampling errors.

A Nonproportional Hazards Weibull Accelerated Failure Time Regression Model

We present a study of risk factors measured in mean before age 50 and subsequent incidence of heart disease over 32 years of follow-up. The data are from the Framingham Heart Study. The standard accelerated failure time model assumes the logarithm of time until an event has a constant dispersion parameter and a location parameter that is a linear function of covariates. Parameters are estimated by maximum likelihood. We reject a standard Weibull model for these data in favor of a model with the dispersion parameter depending on the location parameter. This model suggests that the cumulative hazard ratio for two individuals shrinks towards unity over the follow-up period. Thus, not only the standard Weibull, but also the semiparametric proportional hazards (Cox) model is inadequate for this data. The model improvement appears particularly valuable when estimating the difference in predicted outcome probabilities for two individuals.

The impact of breakthrough clinical trials on survival in population based tumor registries

Three statistical models are developed to study the impact that two breakthrough clinical trials (MOPP for Hodgkin's disease and PVB for disseminated testicular cancer) had on survival in the Connecticut tumor registry and the National Cancer Institute's Surveillance, Epidemiology, and End Results (SEER) registry program. A segmented regression model is used in conjunction with the Cox semi-parametric proportional hazards model, as well as the parametric Weibull and exponential cure models. These models allow us to determine approximately when survival first began to improve dramatically, indicating that improved treatments had become available, and how long it took for survival to level off again indicating that the full population survival impact had been realized. In addition, the degree to which the parametric models fit allows us to determine if the survival improvements occur within a parametric family. Results of the modelling indicate that dissemination took approximately 11 years in Hodgkin's disease while only 3 years in disseminated testicular cancer. In both disease sites survival first broke with prior trends between the time that the breakthrough trial started and its publication, indicating that earlier moderately successful ‘precursor’ trials with combination chemotherapy may have initiated the improved population survival trends. Reasons for the difference in dissemination time in the two cancer sites are examined in order to understand what factors may be responsible for the speed of dissemination and effective utilization of new therapies.

Parametric Inference in a Multiple Renewal Process with Time-Dependent Covariates

Our particular problem arises from a survey of acute respiratory tract infection (ARI), in which 1,312 kindergarten children in Beijing have been followed day by day for 1 year. Six types of infection are considered: upper respiratory tract infection (including influenza), pyogenic tonsillitis, laryngitis, otitis media, bronchitis, and pneumonia. The record of each child consists of not only the types of diseases and the times of illness, but also age, sex, some indices of health, family history, and personal history of respiratory diseases. During the data analysis, in addition, we considered several basic meteorological variables kindly provided by the Beijing Meteorological Observatory. A parametric statistical model based on a renewal process with competing risks and time-dependent covariates is used to analyse these data. Similar models have been considered by several authors, including Kalbfleisch and Prentice ( 1980), Prentice, Williams, and Peterson ( 1981), and Hsieh, Crowley, and Tormey (1983). These authors use reordering of the time scale in order to modify the partial likelihood techniques of the (semiparametric) proportional hazards model. In contrast to this, our parametric model allows a direct approach, avoiding the complicated conditioning argument in the model with time-dependent covariates, enjoying full efficiency (since inference is based on the full likelihood), and performing the analysis directly in chronological time as opposed to the reordered time scales of the above authors. The asymptotic statistical properties are standard for our fully parametric model. The statistical model was proposed in J.-Q. Fang's unpublished Ph.D. dissertation (University of California at Berkeley, 1985). The history of each individual can make several contributions to the full likelihood in the sense of a renewal process; the proportional hazards model is assumed with a general parametric underlying hazard (log-polynomial);